Abstract
Understanding the footprints of chaos in quantum-many-body systems has been under debate for a long time. In this work, we study the echo dynamics of the Sherrington-Kirkpatrick (SK) model with transverse field under effective time reversal. We investigate numerically its quantum and semiclassical dynamics. We explore how chaotic many-body quantum physics can lead to exponential divergence of the echo of observables and we show that it is a result of three requirements: i) the collective nature of the observable, ii) a properly chosen initial state and iii) the existence of a well-defined chaotic semi-classical (large-N) limit. Under these conditions, the echo grows exponentially up to the Ehrenfest time, which scales logarithmically with the number of spins N. In this regime, the echo is well described by the semiclassical (truncated Wigner) approximation. We also discuss a short-range version of the SK model, where the Ehrenfest time does not depend on N and the quantum echo shows only polynomial growth. Our findings provide new insights on scrambling and echo dynamics and how to observe it experimentally.
Highlights
We investigate how chaotic many-body quantum dynamics leads to the exponential divergence of the echo of observables in the transverse Sherrington-Kirkpatrick (SK) spin model with long-range interactions
We have studied the quantum echo dynamics and its exponential divergence in time in the Sherrington-Kirkpatrick model with transverse field
By choosing collective observables and an initial state such that the initial value of the observable is thermodynamically different than its stationary value, the echo grows exponentially, with the same rate of the underlying semi-classical theory
Summary
Understanding how irreversibility arises in classical and quantum systems has been of pivotal importance since the foundations of statistical mechanics [1,2,3,4,5,6]. The SK model can not be mapped to that of a large spin and for this reason there is no simple classical limit, in a way similar to the situation in the SYK model We show both analytically (c.f. Appendix B) and numerically (c.f. Sec. 5.1) that a semiclassical expansion such as the truncated Wigner approximation (TWA) [46,47,48,49] can accurately reproduce both the forward evolution of observables like magnetization (essentially up to infinitely long times) and the echo and the OTOC up to the Ehrenfest time.
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